| L(s) = 1 | − 8.23·3-s + 23.1·7-s + 40.7·9-s − 56.4·11-s + 65.7·13-s − 20.0·17-s + 112.·19-s − 190.·21-s − 140.·23-s − 113.·27-s + 275.·29-s − 1.05·31-s + 464.·33-s + 161.·37-s − 541.·39-s + 414.·41-s + 267.·43-s − 40.1·47-s + 193.·49-s + 165.·51-s − 247.·53-s − 924.·57-s + 19.4·59-s + 26.4·61-s + 944.·63-s − 631.·67-s + 1.16e3·69-s + ⋯ |
| L(s) = 1 | − 1.58·3-s + 1.25·7-s + 1.51·9-s − 1.54·11-s + 1.40·13-s − 0.286·17-s + 1.35·19-s − 1.98·21-s − 1.27·23-s − 0.808·27-s + 1.76·29-s − 0.00612·31-s + 2.44·33-s + 0.719·37-s − 2.22·39-s + 1.57·41-s + 0.948·43-s − 0.124·47-s + 0.564·49-s + 0.453·51-s − 0.641·53-s − 2.14·57-s + 0.0429·59-s + 0.0555·61-s + 1.88·63-s − 1.15·67-s + 2.02·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.389641978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.389641978\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 8.23T + 27T^{2} \) |
| 7 | \( 1 - 23.1T + 343T^{2} \) |
| 11 | \( 1 + 56.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 65.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 20.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 112.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 140.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 275.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 1.05T + 2.97e4T^{2} \) |
| 37 | \( 1 - 161.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 414.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 267.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 40.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + 247.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 19.4T + 2.05e5T^{2} \) |
| 61 | \( 1 - 26.4T + 2.26e5T^{2} \) |
| 67 | \( 1 + 631.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 74.9T + 3.57e5T^{2} \) |
| 73 | \( 1 + 954.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 893.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 782.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 300.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 946.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.612922962525910041514259243460, −7.921073352423765115294756735759, −7.27623151717602435406192447471, −5.99503226926129614553569931669, −5.79065433277526372885946455297, −4.83026523978574078824893351956, −4.31157316580707848743959724613, −2.80922197155597007711739545531, −1.46753740682737992323770701841, −0.64328813409313296563399831958,
0.64328813409313296563399831958, 1.46753740682737992323770701841, 2.80922197155597007711739545531, 4.31157316580707848743959724613, 4.83026523978574078824893351956, 5.79065433277526372885946455297, 5.99503226926129614553569931669, 7.27623151717602435406192447471, 7.921073352423765115294756735759, 8.612922962525910041514259243460
